+\author{Raphaël Couturier \and Lilia Ziane Khodja}
+
+\maketitle
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\begin{abstract}
+In this paper we revist the krylov multisplitting algorithm presented in
+\cite{huang1993krylov} which uses a scalar method to minimize the krylov
+iterations computed by a multisplitting algorithm. Our new algorithm is based on
+a parallel multisplitting algorithm with few blocks of large size using a
+parallel GMRES method inside each block and on a parallel krylov minimization in
+order to improve the convergence. Some large scale experiments with a 3D Poisson
+problem are presented. They show the obtained improvements compared to a
+classical GMRES both in terms of number of iterations and execution times.
+\end{abstract}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\section{Introduction}
+
+Iterative methods are used to solve large sparse linear systems of equations of
+the form $Ax=b$ because they are easier to parallelize than direct ones. Many
+iterative methods have been proposed and adapted by many researchers. When
+solving large linear systems with many cores, iterative methods often suffer
+from scalability problems. This is due to their need for collective
+communications to perform matrix-vector products and reduction operations.
+Preconditionners can be used in order to increase the convergence of iterative
+solvers. However, most of the good preconditionners are not sclalable when
+thousands of cores are used.
+
+
+A completer...
+On ne peut pas parler de tout...\\
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%
+%% BEGIN
+%%%%%%%%%%%%%%%%%%%%%%%
+The key idea of the multisplitting method for solving a large system of linear equations
+$Ax=b$ consists in partitioning the matrix $A$ in $L$ several ways
+\begin{equation}
+A = M_l - N_l,~l\in\{1,\ldots,L\},
+\label{eq01}
+\end{equation}
+where $M_l$ are nonsingular matrices. Then the linear system is solved by iteration based
+on the multisplittings as follows
+\begin{equation}
+x^{k+1}=\displaystyle\sum^L_{l=1} E_l M^{-1}_l (N_l x^k + b),~k=1,2,3,\ldots
+\label{eq02}
+\end{equation}
+where $E_l$ are non-negative and diagonal weighting matrices such that $\sum^L_{l=1}E_l=I$ ($I$ is an identity matrix).
+Thus the convergence of such a method is dependent on the condition
+\begin{equation}
+\rho(\displaystyle\sum^L_{l=1}E_l M^{-1}_l N_l)<1.
+\label{eq03}
+\end{equation}
+
+The advantage of the multisplitting method is that at each iteration $k$ there are $L$ different linear
+systems
+\begin{equation}
+y_l^k=M^{-1}_l N_l x_l^{k-1} + M^{-1}_l b,~l\in\{1,\ldots,L\},
+\label{eq04}
+\end{equation}
+to be solved independently by a direct or an iterative method, where $y_l^k$ is the solution of the local system.
+A multisplitting method using an iterative method for solving the $L$ linear systems is called an inner-outer
+iterative method or a two-stage method. The solution of the global linear system at the iteration $k$ is computed
+as follows
+\begin{equation}
+x^k = \displaystyle\sum^L_{l=1} E_l y_l^k,
+\label{eq05}
+\end{equation}
+In the case where the diagonal weighting matrices $E_l$ have only zero and one factors (i.e. $y_l^k$ are disjoint vectors),
+the multisplitting method is non-overlapping and corresponds to the block Jacobi method.
+%%%%%%%%%%%%%%%%%%%%%%%
+%% END
+%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Related works}
+
+
+A general framework for studying parallel multisplitting has been presented in
+\cite{o1985multi} by O'Leary and White. Convergence conditions are given for the
+most general case. Many authors improved multisplitting algorithms by proposing
+for example a asynchronous version \cite{bru1995parallel} and convergence
+condition \cite{bai1999block,bahi2000asynchronous} in this case or other
+two-stage algorithms~\cite{frommer1992h,bru1995parallel}
+
+In \cite{huang1993krylov}, the authors proposed a parallel multisplitting
+algorithm in which all the tasks except one are devoted to solve a sub-block of
+the splitting and to send their local solution to the first task which is in
+charge to combine the vectors at each iteration. These vectors form a Krylov
+basis for which the first tasks minimize the error function over the basis to
+increase the convergence, then the other tasks receive the update solution until
+convergence of the global system.
+
+
+
+In \cite{couturier2008gremlins}, the authors proposed practical implementations
+of multisplitting algorithms that take benefit from multisplitting algorithms to
+solve large scale linear systems. Inner solvers could be based on scalar direct
+method with the LU method or scalar iterative one with GMRES.
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\section{A two-stage method with a minimization}
+Let $Ax=b$ be a given sparse and large linear system of $n$ equations
+to solve in parallel on $L$ clusters, physically adjacent or geographically
+distant, where $A\in\mathbb{R}^{n\times n}$ is a square and nonsingular
+matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$
+is the right-hand side vector. The multisplitting of this linear system
+is defined as follows:
+\begin{equation}
+\left\{
+\begin{array}{lll}
+A & = & [A_{1}, \ldots, A_{L}]\\
+x & = & [X_{1}, \ldots, X_{L}]\\
+b & = & [B_{1}, \ldots, B_{L}]
+\end{array}
+\right.
+\label{sec03:eq01}
+\end{equation}
+where for all $l\in\{1,\ldots,L\}$ $A_l$ is a rectangular block of size $n_l\times n$
+and $X_l$ and $B_l$ are sub-vectors of size $n_l$, such that $\sum_ln_l=n$. In this
+case, we use a row-by-row splitting without overlapping in such a way that successive
+rows of the sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster.
+So, the multisplitting format of the linear system is defined as follows:
+\begin{equation}
+\forall l\in\{1,\ldots,L\} \mbox{,~} \displaystyle\sum_{i=1}^{l-1}A_{li}X_i + A_{ll}X_l + \displaystyle\sum_{i=l+1}^{L}A_{li}X_i = B_l,
+\label{sec03:eq02}
+\end{equation}
+where $A_{li}$ is a block of size $n_l\times n_i$ of the rectangular matrix $A_l$, $X_i\neq X_l$
+is a sub-vector of size $n_i$ of the solution vector $x$ and $\sum_{i<l}n_i+\sum_{i>l}n_i+n_l=n$,
+for all $i\in\{1,\ldots,l-1,l+1,\ldots,L\}$. Therefore, each cluster $l$ is in charge of solving
+the following spare sub-linear system:
+\begin{equation}
+\left\{
+\begin{array}{l}
+A_{ll}X_l = Y_l \mbox{,~such that}\\
+Y_l = B_l - \displaystyle\sum_{i=1,i\neq l}^{L}A_{li}X_i,
+\end{array}
+\right.
+\label{sec03:eq03}
+\end{equation}
+where the sub-vectors $X_i$ define the data dependencies between the cluster $l$ and other clusters.
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
